## Detailed calculations below:

### Introduction. Fractions

#### A fraction consists of two numbers and a fraction bar: ^{678}/_{206}

#### The number above the bar is the numerator: 678

#### The number below the bar is the denominator: 206

#### The fraction bar means that the two numbers are dividing themselves:

^{678}/_{206} = 678 ÷ 206

#### Divide the numerator by the denominator to get fraction's value:

Value = 678 ÷ 206

### Introduction. Percent

#### 'Percent (%)' means 'out of one hundred':

#### p% = p 'out of one hundred',

#### p% = ^{p}/_{100} = p ÷ 100

### Note:

#### The fraction ^{100}/_{100} = 100 ÷ 100 = 100% = 1

#### Multiply a number by the fraction ^{100}/_{100},

... and its value doesn't change.

## To reduce a fraction, divide both its numerator and denominator by their greatest common factor, GCF.

#### To calculate the greatest common factor, we build the prime factorization of the two numbers.

### Integer numbers prime factorization:

#### Prime Factorization of a number: finding the prime numbers that multiply together to make that number.

#### 678 = 2 × 3 × 113;

678 is not a prime, is a composite number;

#### 206 = 2 × 103;

206 is not a prime, is a composite number;

** Positive integers that are only dividing by themselves and 1 are called prime numbers. A prime number has only two factors: 1 and itself. *

* A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.

### Calculate the greatest (highest) common factor (divisor), gcf, hcf, gcd:

#### Multiply all the common prime factors, by the lowest exponents (if any).

#### gcf, hcf, gcd (678; 206) = 2

## Rewrite the fraction:

### As a mixed number (mixed fraction):

#### A mixed number (mixed fraction): a whole number and a proper fraction, of the same sign.

#### A proper fraction: numerator smaller than denominator.

#### 339 ÷ 103 = 3 and remainder = 30 =>

#### 339 = 3 × 103 + 30 =>

^{339}/_{103} =

^{(3 × 103 + 30)} / _{103} =

^{(3 × 103)} / _{103} + ^{30} / _{103} =

#### 3 + ^{30}/_{103} =

#### 3 ^{30}/_{103}

### As a decimal number:

#### 3 ^{30}/_{103} =

#### 3 + ^{30}/_{103} =

#### 3 + 30 ÷ 103 ≈

#### 3.291262135922 ≈

#### 3.29

### As a percentage:

#### 3.291262135922 =

#### 3.291262135922 × ^{100}/_{100} =

#### ^{329.126213592233}/_{100} =

#### 329.126213592233% ≈

#### 329.13%

#### In other words:

#### 1) Calculate fraction's value.

#### 2) Multiply that number by 100.

#### 3) Add the percent sign % to it.

## Final answer

continued below...